What is the Dot Product?
The dot product (also known as the scalar product or inner product) is a mathematical operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. This operation combines two vectors to produce a scalar quantity.
Algebraic Definition
For two vectors A and B in n-dimensional space:
A·B = Σ(Ai × Bi) for i = 1 to n
In 2D or 3D space, this expands to:
A·B = AxBx + AyBy + (AzBz if in 3D)
This formula demonstrates the commutative property of dot product where A·B = B·A.
Geometric Interpretation
The dot product has an important geometric interpretation. For two vectors A and B, the dot product can be defined as:
A·B = ||A|| × ||B|| × cos(θ)
Where:
- ||A|| is the magnitude (length) of vector A
- ||B|| is the magnitude (length) of vector B
- θ is the angle between the two vectors
The dot product relates to the angle between vectors, which is particularly useful in physics and engineering applications.
The dot product relates to the angle between vectors and their magnitudes
Special Cases
Orthogonal Vectors
When two vectors are perpendicular (θ = 90°), their dot product is zero:
A·B = 0
This property is often used in example problems to test for orthogonality.
Parallel Vectors
When two vectors point in the same direction (θ = 0°), their dot product is the product of their magnitudes:
A·B = ||A|| × ||B||
Opposite Vectors
When two vectors point in opposite directions (θ = 180°), their dot product is the negative product of their magnitudes:
A·B = -||A|| × ||B||
Ready to practice what you've learned? Try our dot product calculator or explore more worked examples.
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